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Fractions

Problems

Problems

Equivalent Fractions

Two fractions are equivalent if they express the same part of a
whole. For example, 2/3 and 4/6 express the same part of a whole.
12/9 and 4/3 are also equivalent.

Two fractions are equivalent if there is a number by which both the
numerator and the denominator of one
fraction can be multiplied or divided to yield the other fraction. For
example, we can multiply the numerator and denominator of 2/3 by 2 to
yield 4/6, and we can divide the numerator and denominator of 12/9 by 3
to yield 4/3.

To find a fraction that is equivalent to another fraction but has a specified
(different) denominator, determine what the old denominator must be multiplied
by to yield the new denominator. Then multiply the old numerator by that same
number. For example, to find a fraction equivalent to 2/9 with a denominator of
45:

1.
9×5 = 45
2.
2×5 = 10
The fraction equivalent to 2/9 is 10/45.

Reducing Fractions

Some fractions, like 6/8, can be written as other fractions with a lower
denominator.
6/8 = 3/4
(Note that 6/8 and 3/4 are equivalent by the above
definition). Others, like 5/8, cannot be written with a lower denominator. 3/4
and 5/8 are said to be in lowest terms because they cannot be reduced
further.

How does one know which fractions can be reduced and which cannot be reduced?
In fractions that can be reduced (fractions not in lowest terms), the numerator
and the denominator share at least one common
factor. In fractions that cannot be reduced
(fractions in lowest terms), the numerator and the denominator share no common
factors; that is, they are relatively prime.

To write a fraction in lowest terms, factor the numerator and the denominator.
Then divide both the numerator and the denominator by the greatest common
factor. For instance, take the following steps to factor
36/126:

Least Common Denominator (LCD)

A common denominator of two numbers is a number that can be divided by the
denominators of both numbers. For example, 1/6 and 4/9 have common denominators
of 18, 36, 54, 72, etc. The least common denominator, or LCD, is the
lowest number that can be divided by the denominators of both numbers.
For example, 18 is the least common denominator of 1/6 and 4/9.

The least common denominator of two fractions is the least common
multiple of their denominators. 18 is the LCM of 6 and
9.

Uses of the Least Common Denominator

The least common denominator is a helpful tool in allowing you to take two
different fractions (ex. 3/4 and 7/11) and write them as equivalent fractions
with the same denominator (ex. 33/44 and 28/44). Such a tool is important in
comparing the size of fractions and because fractions can only be added and
subtracted from each other when they have the same denominator. The first step
in the process is to find the LCD. Then write each fraction as an equivalent
fraction with the LCD as a new denominator, using the two steps detailed in
the section on equivalent fractions.

Example 1: Write 3/14 and 4/21 as fractions with the same
denominator.

II. Write each fraction as an equivalent fraction with the LCD (42) as the new
denominator.

(a)
14×3 = 42
.
3×3 = 9
.
(b)
21×2 = 42
.
4×2 = 8
.

Thus,
3/14 = 9/42
and
4/21 = 8/42
.

Note: The number by which the numerator must be multiplied in Part II
will be the product of the factors of the other denominator that are not factors
of its denominator. Here, 3 was multiplied by 3, which is a factor of 21 but
not of 14, and 4 was multiplied by 2, which is a factor of 14 but not of 21.

Example 2: Write 2/5, 5/12, and 9/8 as fractions with the same
denominator.

Determining Which Fraction is Greater

It is very difficult to tell whether 3/14 is greater or less than 4/21 just by
looking at the two fractions. Here's where writing them with a common
denominator comes in handy. We know that
3/14 = 9/42
and that
4/21 = 8/42
.
Since 9 is greater than 8, 9/42 is greater than 8/42. Thus 3/14 is greater than
4/21.

To determine which of two fractions is greater, write them as fractions with the
same denominator. Then see which new fraction has the larger numerator. This
is the greater fraction.